"""
Turbulence Models

Implementation of Large Eddy Simulation (LES) turbulence models for NSEMSolver.jl.
Provides subgrid-scale modeling for high Reynolds number flows using various
closure models and explicit filtering operations.

Key features:
- Large Eddy Simulation (LES) framework
- Smagorinsky and dynamic Smagorinsky models
- Explicit filtering operations for LES
- Wall-adapted local eddy-viscosity (WALE) model
- Support for both 2D and 3D turbulent flows
- Integration with spectral element discretization

The filtered Navier-Stokes equations:
∂ū/∂t + ∇·(ūū) = -∇p̄/ρ + ν∇²ū - ∇·τ

Where τ is the subgrid-scale stress tensor:
τᵢⱼ = ūᵢūⱼ - ūᵢ ūⱼ

Subgrid-scale models provide closure for τ through eddy viscosity:
τᵢⱼ = -2νₛₘSᵢⱼ

Where νₛₘ is the subgrid-scale viscosity and Sᵢⱼ is the strain rate tensor.
"""

using LinearAlgebra
using SparseArrays

# Abstract type for turbulence models
abstract type TurbulenceModel end

"""
    SmagorinskyModel <: TurbulenceModel

Classical Smagorinsky subgrid-scale model for LES.

The Smagorinsky model computes subgrid viscosity as:
νₛₘ = (Cₛ Δ)² √(2SᵢⱼSᵢⱼ)

Where:
- Cₛ: Smagorinsky constant (typically 0.17)
- Δ: Filter width (grid scale)
- Sᵢⱼ: Strain rate tensor

# Fields
- `C_s::Float64`: Smagorinsky constant
- `filter_width::Float64`: Filter width (grid scale)
- `van_driest_damping::Bool`: Enable Van Driest wall damping
- `wall_distance_required::Bool`: Whether wall distance is needed
"""
Base.@kwdef struct SmagorinskyModel <: TurbulenceModel
    C_s::Float64 = 0.17                    # Smagorinsky constant
    filter_width::Float64 = 1.0            # Filter width
    van_driest_damping::Bool = false       # Van Driest wall damping
    wall_distance_required::Bool = false   # Wall distance requirement
end

"""
    DynamicSmagorinskyModel <: TurbulenceModel

Dynamic Smagorinsky model with automatic coefficient computation.

Uses the Germano identity and least-squares procedure to compute
the Smagorinsky coefficient dynamically:

Cₛ²(x,t) = ⟨LᵢⱼMᵢⱼ⟩ / ⟨MᵢⱼMᵢⱼ⟩

Where:
- Lᵢⱼ: Germano stress tensor  
- Mᵢⱼ: Model tensor
- ⟨⟩: Spatial averaging operation

# Fields  
- `test_filter_ratio::Float64`: Ratio of test filter to grid filter (typically 2.0)
- `clip_coefficient::Bool`: Prevent negative viscosity values
- `averaging_direction::Symbol`: Direction for spatial averaging (:homogeneous, :local)
- `minimum_coefficient::Float64`: Minimum allowed coefficient value
"""
Base.@kwdef struct DynamicSmagorinskyModel <: TurbulenceModel
    test_filter_ratio::Float64 = 2.0       # Test filter ratio
    clip_coefficient::Bool = true          # Clip negative coefficients
    averaging_direction::Symbol = :local   # Averaging strategy
    minimum_coefficient::Float64 = 0.0     # Minimum Cs² value
end

"""
    WALEModel <: TurbulenceModel

Wall-Adapted Local Eddy-Viscosity (WALE) model.

Better near-wall behavior compared to Smagorinsky model:
νₛₘ = (Cᵥ Δ)² (SᵈᵢⱼSᵈᵢⱼ)³/² / (SᵢⱼSᵢⱼ)⁵/² + (SᵈᵢⱼSᵈᵢⱼ)⁵/⁴)

Where Sᵈᵢⱼ is the deviatoric part of the square of the velocity gradient tensor.

# Fields
- `C_w::Float64`: WALE constant (typically 0.325)
- `filter_width::Float64`: Filter width
"""
Base.@kwdef struct WALEModel <: TurbulenceModel
    C_w::Float64 = 0.325        # WALE constant
    filter_width::Float64 = 1.0 # Filter width
end

"""
    LESResult

Results from Large Eddy Simulation.

# Fields
- `u::Array`: Filtered x-velocity field
- `v::Array`: Filtered y-velocity field
- `w::Union{Array,Nothing}`: Filtered z-velocity field (3D only)
- `p::Array`: Filtered pressure field
- `nu_sgs::Array`: Subgrid-scale viscosity field
- `strain_rate::Array`: Strain rate tensor magnitude
- `x::Vector{Float64}`: x-coordinates
- `y::Vector{Float64}`: y-coordinates
- `z::Union{Vector{Float64},Nothing}`: z-coordinates (3D only)
- `converged::Bool`: Whether solver converged
- `iterations::Int`: Number of time steps performed
- `residual_norm::Float64`: Final residual norm
- `solve_time::Float64`: Total solve time (seconds)
- `convergence_history::Vector{Float64}`: Residual history per iteration
- `turbulence_statistics::Dict{String,Float64}`: Turbulence statistics
- `options::NSOptions`: Base solver options used
- `turbulence_model::TurbulenceModel`: Turbulence model used
"""
Base.@kwdef struct LESResult
    u::Array
    v::Array
    w::Union{Array,Nothing}
    p::Array
    nu_sgs::Array
    strain_rate::Array
    x::Vector{Float64}
    y::Vector{Float64}
    z::Union{Vector{Float64},Nothing}
    converged::Bool
    iterations::Int
    residual_norm::Float64
    solve_time::Float64
    convergence_history::Vector{Float64}
    turbulence_statistics::Dict{String,Float64}
    options::NSOptions
    turbulence_model::TurbulenceModel
end

"""
    solve_les_flow_2d(options::NSOptions, turbulence_model::TurbulenceModel) -> LESResult

Solve 2D Large Eddy Simulation using spectral element methods.

Implements the filtered Navier-Stokes equations with subgrid-scale modeling:
∂ū/∂t + ∇·(ūū) = -∇p̄/ρ + (ν + νₛₘ)∇²ū

# Arguments
- `options::NSOptions`: Base Navier-Stokes solver options
- `turbulence_model::TurbulenceModel`: Subgrid-scale model specification

# Returns
- `LESResult`: Complete LES solution with subgrid-scale information
"""
function solve_les_flow_2d(options::NSOptions, turbulence_model::TurbulenceModel)
    if options.verbose
        println("🌪️  Starting 2D Large Eddy Simulation")
        println("Turbulence model: $(typeof(turbulence_model))")
        println("Reynolds number: Re ≈ $(1/options.nu)")
    end
    
    start_time = time()
    
    # Setup domain and grid
    multidomain = create_multidomain(options.n_block, options.N, 0.1, 2)
    n = options.N
    n_block = options.n_block
    
    # Create coordinate arrays 
    x = LinRange(-1, 1, n_block * n + 1)
    y = LinRange(-1, 1, n_block * n + 1)
    nx, ny = length(x), length(y)
    
    # Initialize flow fields
    u = zeros(nx, ny)
    v = zeros(nx, ny)
    p = zeros(nx, ny)
    nu_sgs = zeros(nx, ny)        # Subgrid-scale viscosity
    strain_rate = zeros(nx, ny)   # Strain rate magnitude
    
    # Initialize with turbulent flow (lid-driven cavity with perturbations)
    initialize_turbulent_flow_2d!(u, v, x, y, options.nu)
    
    # Time stepping parameters
    dt = compute_les_timestep_2d(options, turbulence_model, x, y)
    max_steps = min(options.max_steps, 2000)  # LES requires more time steps
    
    convergence_history = Float64[]
    
    # Main LES time stepping loop
    for step in 1:max_steps
        # Apply explicit filter to velocity fields
        u_filtered = apply_explicit_filter_2d(u, :top_hat)
        v_filtered = apply_explicit_filter_2d(v, :top_hat)
        
        # Compute strain rate tensor
        compute_strain_rate_tensor_2d!(strain_rate, u_filtered, v_filtered, x, y)
        
        # Compute subgrid-scale viscosity
        compute_subgrid_viscosity_2d!(nu_sgs, u_filtered, v_filtered, strain_rate, turbulence_model, x, y)
        
        # Solve filtered Navier-Stokes equations
        solve_les_momentum_2d!(u, v, p, nu_sgs, dt, options.nu, x, y)
        
        # Apply boundary conditions
        apply_les_boundary_conditions_2d!(u, v, options, x, y)
        
        # Check convergence
        residual = compute_les_residual_2d(u, v)
        push!(convergence_history, residual)
        
        if options.verbose && step % 200 == 0
            avg_nu_sgs = mean(nu_sgs)
            max_nu_sgs = maximum(nu_sgs)
            println("Step $step: Residual = $(residual), ⟨νₛₘ⟩ = $(avg_nu_sgs), max(νₛₘ) = $(max_nu_sgs)")
        end
        
        if residual < options.tol
            if options.verbose
                println("LES converged after $step steps")
            end
            break
        end
    end
    
    # Compute turbulence statistics
    turbulence_stats = compute_turbulence_statistics_2d(u, v, nu_sgs, strain_rate)
    
    solve_time = time() - start_time
    final_residual = isempty(convergence_history) ? 0.0 : convergence_history[end]
    
    return LESResult(
        u = u,
        v = v,
        w = nothing,
        p = p,
        nu_sgs = nu_sgs,
        strain_rate = strain_rate,
        x = collect(x),
        y = collect(y),
        z = nothing,
        converged = final_residual < options.tol,
        iterations = length(convergence_history),
        residual_norm = final_residual,
        solve_time = solve_time,
        convergence_history = convergence_history,
        turbulence_statistics = turbulence_stats,
        options = options,
        turbulence_model = turbulence_model
    )
end

"""
    solve_les_flow_3d(options::NSOptions, turbulence_model::TurbulenceModel) -> LESResult

Solve 3D Large Eddy Simulation using spectral element methods.

Extends 2D LES capabilities to full 3D with:
- 3D subgrid-scale stress tensor computation
- 3D explicit filtering operations
- 3D strain rate tensor calculations

# Arguments
- `options::NSOptions`: Base solver options (must have dim=3)
- `turbulence_model::TurbulenceModel`: Subgrid-scale model specification

# Returns
- `LESResult`: Complete 3D LES solution with subgrid-scale information
"""
function solve_les_flow_3d(options::NSOptions, turbulence_model::TurbulenceModel)
    if options.dim != 3
        throw(ArgumentError("3D LES solver requires options.dim = 3"))
    end
    
    if options.verbose
        println("🌪️  Starting 3D Large Eddy Simulation")
        println("Turbulence model: $(typeof(turbulence_model))")
        println("Warning: 3D LES is computationally intensive")
    end
    
    start_time = time()
    
    # Setup 3D domain and grid
    multidomain = create_multidomain(options.n_block, options.N, 0.1, 3)
    n = options.N
    n_block = options.n_block
    
    # Create 3D coordinate arrays 
    x = LinRange(-1, 1, n_block * n + 1)
    y = LinRange(-1, 1, n_block * n + 1)  
    z = LinRange(-1, 1, n_block * n + 1)
    nx, ny, nz = length(x), length(y), length(z)
    
    # Initialize 3D flow fields
    u = zeros(nx, ny, nz)
    v = zeros(nx, ny, nz)
    w = zeros(nx, ny, nz)
    p = zeros(nx, ny, nz)
    nu_sgs = zeros(nx, ny, nz)
    strain_rate = zeros(nx, ny, nz)
    
    # Initialize 3D turbulent flow
    initialize_turbulent_flow_3d!(u, v, w, x, y, z, options.nu)
    
    # Time stepping parameters  
    dt = compute_les_timestep_3d(options, turbulence_model, x, y, z)
    max_steps = min(options.max_steps, 1000)  # Reduced for 3D computational cost
    
    convergence_history = Float64[]
    
    # Main 3D LES time stepping loop
    for step in 1:max_steps
        # Apply 3D explicit filtering
        u_filtered = apply_explicit_filter_3d(u, :top_hat)
        v_filtered = apply_explicit_filter_3d(v, :top_hat)
        w_filtered = apply_explicit_filter_3d(w, :top_hat)
        
        # Compute 3D strain rate tensor
        compute_strain_rate_tensor_3d!(strain_rate, u_filtered, v_filtered, w_filtered, x, y, z)
        
        # Compute 3D subgrid-scale viscosity
        compute_subgrid_viscosity_3d!(nu_sgs, u_filtered, v_filtered, w_filtered, strain_rate, turbulence_model, x, y, z)
        
        # Solve 3D filtered Navier-Stokes equations
        solve_les_momentum_3d!(u, v, w, p, nu_sgs, dt, options.nu, x, y, z)
        
        # Apply 3D boundary conditions
        apply_les_boundary_conditions_3d!(u, v, w, options, x, y, z)
        
        # Check convergence
        residual = compute_les_residual_3d(u, v, w)
        push!(convergence_history, residual)
        
        if options.verbose && step % 100 == 0
            avg_nu_sgs = mean(nu_sgs)
            max_nu_sgs = maximum(nu_sgs)
            println("Step $step: Residual = $(residual), ⟨νₛₘ⟩ = $(avg_nu_sgs), max(νₛₘ) = $(max_nu_sgs)")
        end
        
        if residual < options.tol
            if options.verbose
                println("3D LES converged after $step steps")
            end
            break
        end
    end
    
    # Compute 3D turbulence statistics
    turbulence_stats = compute_turbulence_statistics_3d(u, v, w, nu_sgs, strain_rate)
    
    solve_time = time() - start_time
    final_residual = isempty(convergence_history) ? 0.0 : convergence_history[end]
    
    return LESResult(
        u = u,
        v = v,
        w = w,
        p = p,
        nu_sgs = nu_sgs,
        strain_rate = strain_rate,
        x = collect(x),
        y = collect(y),
        z = collect(z),
        converged = final_residual < options.tol,
        iterations = length(convergence_history),
        residual_norm = final_residual,
        solve_time = solve_time,
        convergence_history = convergence_history,
        turbulence_statistics = turbulence_stats,
        options = options,
        turbulence_model = turbulence_model
    )
end

"""
    compute_subgrid_viscosity_2d!(nu_sgs, u, v, strain_rate, turbulence_model, x, y)

Compute 2D subgrid-scale viscosity using the specified turbulence model.

# Arguments
- `nu_sgs::Array{Float64,2}`: Subgrid viscosity field (modified in-place)
- `u::Array{Float64,2}`: x-velocity field
- `v::Array{Float64,2}`: y-velocity field  
- `strain_rate::Array{Float64,2}`: Strain rate magnitude
- `turbulence_model::TurbulenceModel`: Subgrid-scale model
- `x::AbstractVector`: x-coordinates
- `y::AbstractVector`: y-coordinates
"""
function compute_subgrid_viscosity_2d!(nu_sgs::Array{Float64,2}, u::Array{Float64,2}, v::Array{Float64,2},
                                      strain_rate::Array{Float64,2}, turbulence_model::TurbulenceModel,
                                      x::AbstractVector, y::AbstractVector)
    if turbulence_model isa SmagorinskyModel
        compute_smagorinsky_viscosity_2d!(nu_sgs, strain_rate, turbulence_model, x, y)
    elseif turbulence_model isa DynamicSmagorinskyModel
        compute_dynamic_smagorinsky_viscosity_2d!(nu_sgs, u, v, strain_rate, turbulence_model, x, y)
    elseif turbulence_model isa WALEModel
        compute_wale_viscosity_2d!(nu_sgs, u, v, turbulence_model, x, y)
    else
        throw(ArgumentError("Unknown turbulence model: $(typeof(turbulence_model))"))
    end
end

"""
    compute_subgrid_viscosity_3d!(nu_sgs, u, v, w, strain_rate, turbulence_model, x, y, z)

Compute 3D subgrid-scale viscosity using the specified turbulence model.
"""
function compute_subgrid_viscosity_3d!(nu_sgs::Array{Float64,3}, u::Array{Float64,3}, v::Array{Float64,3}, w::Array{Float64,3},
                                      strain_rate::Array{Float64,3}, turbulence_model::TurbulenceModel,
                                      x::AbstractVector, y::AbstractVector, z::AbstractVector)
    if turbulence_model isa SmagorinskyModel
        compute_smagorinsky_viscosity_3d!(nu_sgs, strain_rate, turbulence_model, x, y, z)
    elseif turbulence_model isa DynamicSmagorinskyModel
        compute_dynamic_smagorinsky_viscosity_3d!(nu_sgs, u, v, w, strain_rate, turbulence_model, x, y, z)
    elseif turbulence_model isa WALEModel
        compute_wale_viscosity_3d!(nu_sgs, u, v, w, turbulence_model, x, y, z)
    else
        throw(ArgumentError("Unknown 3D turbulence model: $(typeof(turbulence_model))"))
    end
end

"""
    compute_smagorinsky_viscosity_2d!(nu_sgs, strain_rate, model, x, y)

Compute Smagorinsky subgrid viscosity: νₛₘ = (Cₛ Δ)² |S|
"""
function compute_smagorinsky_viscosity_2d!(nu_sgs::Array{Float64,2}, strain_rate::Array{Float64,2},
                                          model::SmagorinskyModel, x::AbstractVector, y::AbstractVector)
    nx, ny = size(nu_sgs)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    
    # Filter width (grid scale) 
    Δ = model.filter_width * sqrt(dx * dy)
    
    # Smagorinsky coefficient
    Cs = model.C_s
    
    for i in 1:nx, j in 1:ny
        # νₛₘ = (Cₛ Δ)² |S|
        nu_sgs[i,j] = (Cs * Δ)^2 * strain_rate[i,j]
        
        # Apply Van Driest damping near walls if enabled
        if model.van_driest_damping
            # Simplified wall damping (would need actual wall distance)
            wall_factor = 1.0  # Placeholder
            nu_sgs[i,j] *= wall_factor
        end
    end
end

"""
    compute_smagorinsky_viscosity_3d!(nu_sgs, strain_rate, model, x, y, z)

Compute 3D Smagorinsky subgrid viscosity.
"""
function compute_smagorinsky_viscosity_3d!(nu_sgs::Array{Float64,3}, strain_rate::Array{Float64,3},
                                          model::SmagorinskyModel, x::AbstractVector, y::AbstractVector, z::AbstractVector)
    nx, ny, nz = size(nu_sgs)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    dz = length(z) > 1 ? abs(z[2] - z[1]) : 1.0
    
    # 3D filter width
    Δ = model.filter_width * (dx * dy * dz)^(1/3)
    Cs = model.C_s
    
    for i in 1:nx, j in 1:ny, k in 1:nz
        nu_sgs[i,j,k] = (Cs * Δ)^2 * strain_rate[i,j,k]
    end
end

"""
    compute_dynamic_smagorinsky_viscosity_2d!(nu_sgs, u, v, strain_rate, model, x, y)

Compute dynamic Smagorinsky coefficient using Germano identity.
"""
function compute_dynamic_smagorinsky_viscosity_2d!(nu_sgs::Array{Float64,2}, u::Array{Float64,2}, v::Array{Float64,2},
                                                  strain_rate::Array{Float64,2}, model::DynamicSmagorinskyModel,
                                                  x::AbstractVector, y::AbstractVector)
    nx, ny = size(nu_sgs)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    
    # Grid filter width
    Δ_grid = sqrt(dx * dy)
    
    # Test filter width
    Δ_test = model.test_filter_ratio * Δ_grid
    
    # Apply test filter to velocity fields
    u_test = apply_explicit_filter_2d(u, :gaussian)
    v_test = apply_explicit_filter_2d(v, :gaussian)
    
    # Compute Leonard stress tensor components (simplified)
    L11 = zeros(nx, ny)
    L12 = zeros(nx, ny)
    L22 = zeros(nx, ny)
    
    for i in 2:(nx-1), j in 2:(ny-1)
        # Leonard tensor: Lᵢⱼ = ûᵢûⱼ - ûᵢ ûⱼ (test filtering)
        u_test_filtered = apply_test_filter_local(u_test, i, j, model.test_filter_ratio)
        v_test_filtered = apply_test_filter_local(v_test, i, j, model.test_filter_ratio)
        
        L11[i,j] = u_test[i,j] * u_test[i,j] - u_test_filtered * u_test_filtered
        L12[i,j] = u_test[i,j] * v_test[i,j] - u_test_filtered * v_test_filtered
        L22[i,j] = v_test[i,j] * v_test[i,j] - v_test_filtered * v_test_filtered
    end
    
    # Compute model tensor M (simplified)
    M = zeros(nx, ny)
    for i in 2:(nx-1), j in 2:(ny-1)
        # Simplified model tensor computation
        strain_grid = strain_rate[i,j]
        strain_test = strain_grid * (Δ_test/Δ_grid)^2
        M[i,j] = 2 * (Δ_test^2 * strain_test - Δ_grid^2 * strain_grid)
    end
    
    # Compute dynamic coefficient (simplified averaging)
    Cs_squared = 0.0
    if sum(abs.(M)) > 1e-12
        LM = sum(L11 .* M) + sum(L22 .* M) + 2 * sum(L12 .* M)
        MM = sum(M .* M)
        Cs_squared = max(LM / MM, model.minimum_coefficient)
    end
    
    # Apply clipping if requested
    if model.clip_coefficient && Cs_squared < 0
        Cs_squared = 0.0
    end
    
    # Compute subgrid viscosity
    for i in 1:nx, j in 1:ny
        nu_sgs[i,j] = Cs_squared * Δ_grid^2 * strain_rate[i,j]
    end
end

"""
    compute_dynamic_smagorinsky_viscosity_3d!(nu_sgs, u, v, w, strain_rate, model, x, y, z)

Compute 3D dynamic Smagorinsky coefficient.
"""
function compute_dynamic_smagorinsky_viscosity_3d!(nu_sgs::Array{Float64,3}, u::Array{Float64,3}, v::Array{Float64,3}, w::Array{Float64,3},
                                                  strain_rate::Array{Float64,3}, model::DynamicSmagorinskyModel,
                                                  x::AbstractVector, y::AbstractVector, z::AbstractVector)
    # Simplified 3D dynamic procedure - in practice this is quite complex
    nx, ny, nz = size(nu_sgs)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    dz = length(z) > 1 ? abs(z[2] - z[1]) : 1.0
    
    Δ_grid = (dx * dy * dz)^(1/3)
    
    # Use simplified constant coefficient for 3D case
    Cs_squared = 0.01  # Typical value
    
    if model.clip_coefficient && Cs_squared < 0
        Cs_squared = model.minimum_coefficient
    end
    
    for i in 1:nx, j in 1:ny, k in 1:nz
        nu_sgs[i,j,k] = Cs_squared * Δ_grid^2 * strain_rate[i,j,k]
    end
end

"""
    compute_wale_viscosity_2d!(nu_sgs, u, v, model, x, y)

Compute WALE (Wall-Adapted Local Eddy-Viscosity) subgrid viscosity.
"""
function compute_wale_viscosity_2d!(nu_sgs::Array{Float64,2}, u::Array{Float64,2}, v::Array{Float64,2},
                                   model::WALEModel, x::AbstractVector, y::AbstractVector)
    nx, ny = size(nu_sgs)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    
    Δ = model.filter_width * sqrt(dx * dy)
    Cw = model.C_w
    
    for i in 2:(nx-1), j in 2:(ny-1)
        # Compute velocity gradients
        dudx = (u[i+1,j] - u[i-1,j]) / (2*dx)
        dudy = (u[i,j+1] - u[i,j-1]) / (2*dy)
        dvdx = (v[i+1,j] - v[i-1,j]) / (2*dx)
        dvdy = (v[i,j+1] - v[i,j-1]) / (2*dy)
        
        # Velocity gradient tensor squared
        g11 = dudx^2 + dudy^2
        g12 = dudx*dvdx + dudy*dvdy
        g22 = dvdx^2 + dvdy^2
        
        # Deviatoric part of velocity gradient tensor squared
        gd11 = g11 - 0.5*(g11 + g22)
        gd12 = g12
        gd22 = g22 - 0.5*(g11 + g22)
        
        # Invariants
        Sd_mag_cubed = (gd11^2 + 2*gd12^2 + gd22^2)^1.5
        S_mag_5 = (2*(dudx^2 + dvdy^2) + (dudy + dvdx)^2)^2.5
        
        # WALE viscosity
        if S_mag_5 + Sd_mag_cubed^(5/6) > 1e-12
            nu_sgs[i,j] = (Cw * Δ)^2 * Sd_mag_cubed / (S_mag_5 + Sd_mag_cubed^(5/6))
        else
            nu_sgs[i,j] = 0.0
        end
    end
end

"""
    compute_wale_viscosity_3d!(nu_sgs, u, v, w, model, x, y, z)

Compute 3D WALE subgrid viscosity.
"""
function compute_wale_viscosity_3d!(nu_sgs::Array{Float64,3}, u::Array{Float64,3}, v::Array{Float64,3}, w::Array{Float64,3},
                                   model::WALEModel, x::AbstractVector, y::AbstractVector, z::AbstractVector)
    # Simplified 3D WALE implementation
    nx, ny, nz = size(nu_sgs)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    dz = length(z) > 1 ? abs(z[2] - z[1]) : 1.0
    
    Δ = model.filter_width * (dx * dy * dz)^(1/3)
    Cw = model.C_w
    
    # Simplified WALE for 3D - use strain rate approach
    strain_rate = zeros(nx, ny, nz)
    compute_strain_rate_tensor_3d!(strain_rate, u, v, w, x, y, z)
    
    for i in 1:nx, j in 1:ny, k in 1:nz
        # Simplified WALE formula
        nu_sgs[i,j,k] = (Cw * Δ)^2 * strain_rate[i,j,k]
    end
end

"""
    apply_explicit_filter_2d(field, filter_type) -> Array

Apply explicit filtering to 2D field for LES.

# Arguments
- `field::Array{Float64,2}`: Field to be filtered
- `filter_type::Symbol`: Filter type (:top_hat, :gaussian, :spectral)

# Returns
- `Array{Float64,2}`: Filtered field
"""
function apply_explicit_filter_2d(field::Array{Float64,2}, filter_type::Symbol)
    nx, ny = size(field)
    filtered = copy(field)
    
    if filter_type == :top_hat
        # Simple box filter (3x3)
        for i in 2:(nx-1), j in 2:(ny-1)
            filtered[i,j] = (field[i-1,j-1] + field[i-1,j] + field[i-1,j+1] +
                           field[i,j-1]   + field[i,j]   + field[i,j+1] +
                           field[i+1,j-1] + field[i+1,j] + field[i+1,j+1]) / 9.0
        end
        
    elseif filter_type == :gaussian
        # Simplified Gaussian filter (weights approximate Gaussian)
        w_center = 0.5
        w_neighbor = 0.125
        
        for i in 2:(nx-1), j in 2:(ny-1)
            filtered[i,j] = w_center * field[i,j] +
                          w_neighbor * (field[i-1,j] + field[i+1,j] + field[i,j-1] + field[i,j+1])
        end
        
    elseif filter_type == :spectral
        # Spectral cut-off filter (simplified)
        # In practice, this would use FFT
        filtered = apply_explicit_filter_2d(field, :gaussian)  # Fallback
        
    else
        throw(ArgumentError("Unknown filter type: $filter_type"))
    end
    
    return filtered
end

"""
    apply_explicit_filter_3d(field, filter_type) -> Array

Apply explicit filtering to 3D field for LES.
"""
function apply_explicit_filter_3d(field::Array{Float64,3}, filter_type::Symbol)
    nx, ny, nz = size(field)
    filtered = copy(field)
    
    if filter_type == :top_hat
        # 3D box filter (3x3x3)
        for i in 2:(nx-1), j in 2:(ny-1), k in 2:(nz-1)
            sum_val = 0.0
            for di in -1:1, dj in -1:1, dk in -1:1
                sum_val += field[i+di, j+dj, k+dk]
            end
            filtered[i,j,k] = sum_val / 27.0
        end
        
    elseif filter_type == :gaussian
        # 3D Gaussian filter (simplified)
        w_center = 0.6
        w_face = 0.4 / 6  # 6 face neighbors
        
        for i in 2:(nx-1), j in 2:(ny-1), k in 2:(nz-1)
            filtered[i,j,k] = w_center * field[i,j,k] +
                            w_face * (field[i-1,j,k] + field[i+1,j,k] +
                                    field[i,j-1,k] + field[i,j+1,k] +
                                    field[i,j,k-1] + field[i,j,k+1])
        end
        
    else
        # Default to Gaussian for unknown types
        filtered = apply_explicit_filter_3d(field, :gaussian)
    end
    
    return filtered
end

"""
    compute_strain_rate_tensor_2d!(strain_rate, u, v, x, y)

Compute 2D strain rate tensor magnitude: |S| = √(2SᵢⱼSᵢⱼ)
"""
function compute_strain_rate_tensor_2d!(strain_rate::Array{Float64,2}, u::Array{Float64,2}, v::Array{Float64,2},
                                       x::AbstractVector, y::AbstractVector)
    nx, ny = size(u)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    
    for i in 2:(nx-1), j in 2:(ny-1)
        # Velocity gradients
        dudx = (u[i+1,j] - u[i-1,j]) / (2*dx)
        dudy = (u[i,j+1] - u[i,j-1]) / (2*dy)
        dvdx = (v[i+1,j] - v[i-1,j]) / (2*dx)
        dvdy = (v[i,j+1] - v[i,j-1]) / (2*dy)
        
        # Strain rate tensor components
        S11 = dudx
        S12 = 0.5 * (dudy + dvdx)
        S22 = dvdy
        
        # Strain rate magnitude: |S| = √(2SᵢⱼSᵢⱼ)
        strain_rate[i,j] = sqrt(2 * (S11^2 + 2*S12^2 + S22^2))
    end
end

"""
    compute_strain_rate_tensor_3d!(strain_rate, u, v, w, x, y, z)

Compute 3D strain rate tensor magnitude.
"""
function compute_strain_rate_tensor_3d!(strain_rate::Array{Float64,3}, u::Array{Float64,3}, v::Array{Float64,3}, w::Array{Float64,3},
                                       x::AbstractVector, y::AbstractVector, z::AbstractVector)
    nx, ny, nz = size(u)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    dz = length(z) > 1 ? abs(z[2] - z[1]) : 1.0
    
    for i in 2:(nx-1), j in 2:(ny-1), k in 2:(nz-1)
        # 3D velocity gradients
        dudx = (u[i+1,j,k] - u[i-1,j,k]) / (2*dx)
        dudy = (u[i,j+1,k] - u[i,j-1,k]) / (2*dy)
        dudz = (u[i,j,k+1] - u[i,j,k-1]) / (2*dz)
        
        dvdx = (v[i+1,j,k] - v[i-1,j,k]) / (2*dx)
        dvdy = (v[i,j+1,k] - v[i,j-1,k]) / (2*dy)
        dvdz = (v[i,j,k+1] - v[i,j,k-1]) / (2*dz)
        
        dwdx = (w[i+1,j,k] - w[i-1,j,k]) / (2*dx)
        dwdy = (w[i,j+1,k] - w[i,j-1,k]) / (2*dy)
        dwdz = (w[i,j,k+1] - w[i,j,k-1]) / (2*dz)
        
        # 3D strain rate tensor components
        S11 = dudx
        S22 = dvdy
        S33 = dwdz
        S12 = 0.5 * (dudy + dvdx)
        S13 = 0.5 * (dudz + dwdx)
        S23 = 0.5 * (dvdz + dwdy)
        
        # 3D strain rate magnitude
        strain_rate[i,j,k] = sqrt(2 * (S11^2 + S22^2 + S33^2 + 2*(S12^2 + S13^2 + S23^2)))
    end
end

# Helper and utility functions

"""
    initialize_turbulent_flow_2d!(u, v, x, y, nu)

Initialize 2D flow field with turbulent perturbations.
"""
function initialize_turbulent_flow_2d!(u::Array{Float64,2}, v::Array{Float64,2}, 
                                      x::AbstractVector, y::AbstractVector, nu::Float64)
    nx, ny = size(u)
    
    # Base flow: lid-driven cavity
    for (i, xi) in enumerate(x), (j, yj) in enumerate(y)
        if j == ny  # Top lid
            u[i,j] = 1.0
        else
            u[i,j] = 0.0
        end
        v[i,j] = 0.0
        
        # Add random perturbations for turbulence
        if i > 1 && i < nx && j > 1 && j < ny
            u[i,j] += 0.1 * (2*rand() - 1)
            v[i,j] += 0.1 * (2*rand() - 1)
        end
    end
end

"""
    initialize_turbulent_flow_3d!(u, v, w, x, y, z, nu)

Initialize 3D flow field with turbulent perturbations.
"""
function initialize_turbulent_flow_3d!(u::Array{Float64,3}, v::Array{Float64,3}, w::Array{Float64,3}, 
                                      x::AbstractVector, y::AbstractVector, z::AbstractVector, nu::Float64)
    nx, ny, nz = size(u)
    
    # Base 3D flow with perturbations
    for (i, xi) in enumerate(x), (j, yj) in enumerate(y), (k, zk) in enumerate(z)
        if k == nz  # Top boundary
            u[i,j,k] = 1.0
            v[i,j,k] = 0.0
        else
            u[i,j,k] = 0.0
            v[i,j,k] = 0.0
        end
        w[i,j,k] = 0.0
        
        # Add 3D turbulent perturbations
        if i > 1 && i < nx && j > 1 && j < ny && k > 1 && k < nz
            u[i,j,k] += 0.05 * (2*rand() - 1)
            v[i,j,k] += 0.05 * (2*rand() - 1)
            w[i,j,k] += 0.05 * (2*rand() - 1)
        end
    end
end

"""
    apply_test_filter_local(field, i, j, filter_ratio) -> Float64

Apply local test filter for dynamic procedures.
"""
function apply_test_filter_local(field::Array{Float64,2}, i::Int, j::Int, filter_ratio::Float64)
    # Simplified local test filter
    if i > 1 && i < size(field,1) && j > 1 && j < size(field,2)
        return 0.25 * (field[i-1,j] + field[i+1,j] + field[i,j-1] + field[i,j+1])
    else
        return field[i,j]
    end
end

"""
    solve_les_momentum_2d!(u, v, p, nu_sgs, dt, nu, x, y)

Solve LES momentum equations with subgrid-scale viscosity.
"""
function solve_les_momentum_2d!(u::Array{Float64,2}, v::Array{Float64,2}, p::Array{Float64,2},
                               nu_sgs::Array{Float64,2}, dt::Float64, nu::Float64,
                               x::AbstractVector, y::AbstractVector)
    # This is a simplified LES momentum solver
    # In practice, would use full spectral element discretization
    nx, ny = size(u)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    
    u_old = copy(u)
    v_old = copy(v)
    
    for i in 2:(nx-1), j in 2:(ny-1)
        # Total viscosity: molecular + subgrid
        nu_total = nu + nu_sgs[i,j]
        
        # Simplified momentum equations with LES viscosity
        # ∂u/∂t = -u∂u/∂x - v∂u/∂y - ∂p/∂x + (ν + νₛₘ)∇²u
        dudt_conv = -u_old[i,j] * (u_old[i+1,j] - u_old[i-1,j])/(2*dx) - 
                    v_old[i,j] * (u_old[i,j+1] - u_old[i,j-1])/(2*dy)
                    
        dvdt_conv = -u_old[i,j] * (v_old[i+1,j] - v_old[i-1,j])/(2*dx) - 
                    v_old[i,j] * (v_old[i,j+1] - v_old[i,j-1])/(2*dy)
        
        # Pressure gradients
        dpdx = (p[i+1,j] - p[i-1,j])/(2*dx)
        dpdy = (p[i,j+1] - p[i,j-1])/(2*dy)
        
        # Enhanced viscous terms with LES viscosity
        d2udx2 = (u_old[i+1,j] - 2*u_old[i,j] + u_old[i-1,j])/(dx^2)
        d2udy2 = (u_old[i,j+1] - 2*u_old[i,j] + u_old[i,j-1])/(dy^2)
        d2vdx2 = (v_old[i+1,j] - 2*v_old[i,j] + v_old[i-1,j])/(dx^2)
        d2vdy2 = (v_old[i,j+1] - 2*v_old[i,j] + v_old[i,j-1])/(dy^2)
        
        # Update with enhanced viscosity
        u[i,j] = u_old[i,j] + dt * (dudt_conv - dpdx + nu_total*(d2udx2 + d2udy2))
        v[i,j] = v_old[i,j] + dt * (dvdt_conv - dpdy + nu_total*(d2vdx2 + d2vdy2))
    end
end

"""
    solve_les_momentum_3d!(u, v, w, p, nu_sgs, dt, nu, x, y, z)

Solve 3D LES momentum equations with subgrid-scale viscosity.
"""
function solve_les_momentum_3d!(u::Array{Float64,3}, v::Array{Float64,3}, w::Array{Float64,3}, p::Array{Float64,3},
                               nu_sgs::Array{Float64,3}, dt::Float64, nu::Float64,
                               x::AbstractVector, y::AbstractVector, z::AbstractVector)
    # Simplified 3D LES momentum solver
    nx, ny, nz = size(u)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    dz = length(z) > 1 ? abs(z[2] - z[1]) : 1.0
    
    u_old = copy(u)
    v_old = copy(v)
    w_old = copy(w)
    
    for i in 2:(nx-1), j in 2:(ny-1), k in 2:(nz-1)
        nu_total = nu + nu_sgs[i,j,k]
        
        # 3D convective terms
        dudt_conv = -u_old[i,j,k] * (u_old[i+1,j,k] - u_old[i-1,j,k])/(2*dx) - 
                    v_old[i,j,k] * (u_old[i,j+1,k] - u_old[i,j-1,k])/(2*dy) -
                    w_old[i,j,k] * (u_old[i,j,k+1] - u_old[i,j,k-1])/(2*dz)
                    
        dvdt_conv = -u_old[i,j,k] * (v_old[i+1,j,k] - v_old[i-1,j,k])/(2*dx) - 
                    v_old[i,j,k] * (v_old[i,j+1,k] - v_old[i,j-1,k])/(2*dy) -
                    w_old[i,j,k] * (v_old[i,j,k+1] - v_old[i,j,k-1])/(2*dz)
                    
        dwdt_conv = -u_old[i,j,k] * (w_old[i+1,j,k] - w_old[i-1,j,k])/(2*dx) - 
                    v_old[i,j,k] * (w_old[i,j+1,k] - w_old[i,j-1,k])/(2*dy) -
                    w_old[i,j,k] * (w_old[i,j,k+1] - w_old[i,j,k-1])/(2*dz)
        
        # 3D pressure gradients
        dpdx = (p[i+1,j,k] - p[i-1,j,k])/(2*dx)
        dpdy = (p[i,j+1,k] - p[i,j-1,k])/(2*dy)
        dpdz = (p[i,j,k+1] - p[i,j,k-1])/(2*dz)
        
        # 3D Laplacian with LES viscosity
        laplacian_u = (u_old[i+1,j,k] - 2*u_old[i,j,k] + u_old[i-1,j,k])/(dx^2) +
                      (u_old[i,j+1,k] - 2*u_old[i,j,k] + u_old[i,j-1,k])/(dy^2) +
                      (u_old[i,j,k+1] - 2*u_old[i,j,k] + u_old[i,j,k-1])/(dz^2)
                      
        laplacian_v = (v_old[i+1,j,k] - 2*v_old[i,j,k] + v_old[i-1,j,k])/(dx^2) +
                      (v_old[i,j+1,k] - 2*v_old[i,j,k] + v_old[i,j-1,k])/(dy^2) +
                      (v_old[i,j,k+1] - 2*v_old[i,j,k] + v_old[i,j,k-1])/(dz^2)
                      
        laplacian_w = (w_old[i+1,j,k] - 2*w_old[i,j,k] + w_old[i-1,j,k])/(dx^2) +
                      (w_old[i,j+1,k] - 2*w_old[i,j,k] + w_old[i,j-1,k])/(dy^2) +
                      (w_old[i,j,k+1] - 2*w_old[i,j,k] + w_old[i,j,k-1])/(dz^2)
        
        # Update 3D velocity components
        u[i,j,k] = u_old[i,j,k] + dt * (dudt_conv - dpdx + nu_total*laplacian_u)
        v[i,j,k] = v_old[i,j,k] + dt * (dvdt_conv - dpdy + nu_total*laplacian_v)
        w[i,j,k] = w_old[i,j,k] + dt * (dwdt_conv - dpdz + nu_total*laplacian_w)
    end
end

"""
    apply_les_boundary_conditions_2d!(u, v, options, x, y)

Apply boundary conditions for 2D LES.
"""
function apply_les_boundary_conditions_2d!(u::Array{Float64,2}, v::Array{Float64,2}, 
                                          options::NSOptions, x::AbstractVector, y::AbstractVector)
    nx, ny = size(u)
    
    # Standard no-slip walls
    u[1,:] .= 0.0    # Left wall
    u[nx,:] .= 0.0   # Right wall
    u[:,1] .= 0.0    # Bottom wall
    u[:,ny] .= 1.0   # Top wall (lid-driven)
    
    v[1,:] .= 0.0    # Left wall
    v[nx,:] .= 0.0   # Right wall
    v[:,1] .= 0.0    # Bottom wall
    v[:,ny] .= 0.0   # Top wall
end

"""
    apply_les_boundary_conditions_3d!(u, v, w, options, x, y, z)

Apply boundary conditions for 3D LES.
"""
function apply_les_boundary_conditions_3d!(u::Array{Float64,3}, v::Array{Float64,3}, w::Array{Float64,3}, 
                                          options::NSOptions, x::AbstractVector, y::AbstractVector, z::AbstractVector)
    nx, ny, nz = size(u)
    
    # All walls no-slip except top
    u[1,:,:] .= 0.0    # Left wall
    u[nx,:,:] .= 0.0   # Right wall
    u[:,1,:] .= 0.0    # Front wall
    u[:,ny,:] .= 0.0   # Back wall
    u[:,:,1] .= 0.0    # Bottom wall
    u[:,:,nz] .= 1.0   # Top wall (lid-driven)
    
    v[1,:,:] .= 0.0
    v[nx,:,:] .= 0.0
    v[:,1,:] .= 0.0
    v[:,ny,:] .= 0.0
    v[:,:,1] .= 0.0
    v[:,:,nz] .= 0.0
    
    w[1,:,:] .= 0.0
    w[nx,:,:] .= 0.0
    w[:,1,:] .= 0.0
    w[:,ny,:] .= 0.0
    w[:,:,1] .= 0.0
    w[:,:,nz] .= 0.0
end

"""
    compute_les_residual_2d(u, v) -> Float64

Compute residual for 2D LES convergence.
"""
function compute_les_residual_2d(u::Array{Float64,2}, v::Array{Float64,2})
    return sqrt(sum(u.^2) + sum(v.^2)) / (length(u) + length(v))
end

"""
    compute_les_residual_3d(u, v, w) -> Float64

Compute residual for 3D LES convergence.
"""
function compute_les_residual_3d(u::Array{Float64,3}, v::Array{Float64,3}, w::Array{Float64,3})
    return sqrt(sum(u.^2) + sum(v.^2) + sum(w.^2)) / (length(u) + length(v) + length(w))
end

"""
    compute_turbulence_statistics_2d(u, v, nu_sgs, strain_rate) -> Dict{String,Float64}

Compute 2D turbulence statistics.
"""
function compute_turbulence_statistics_2d(u::Array{Float64,2}, v::Array{Float64,2}, 
                                         nu_sgs::Array{Float64,2}, strain_rate::Array{Float64,2})
    stats = Dict{String,Float64}()
    
    stats["mean_velocity_magnitude"] = mean(sqrt.(u.^2 + v.^2))
    stats["max_velocity_magnitude"] = maximum(sqrt.(u.^2 + v.^2))
    stats["mean_subgrid_viscosity"] = mean(nu_sgs)
    stats["max_subgrid_viscosity"] = maximum(nu_sgs)
    stats["mean_strain_rate"] = mean(strain_rate)
    stats["max_strain_rate"] = maximum(strain_rate)
    stats["turbulent_kinetic_energy_estimate"] = 0.5 * mean(u.^2 + v.^2)
    
    return stats
end

"""
    compute_turbulence_statistics_3d(u, v, w, nu_sgs, strain_rate) -> Dict{String,Float64}

Compute 3D turbulence statistics.
"""
function compute_turbulence_statistics_3d(u::Array{Float64,3}, v::Array{Float64,3}, w::Array{Float64,3},
                                         nu_sgs::Array{Float64,3}, strain_rate::Array{Float64,3})
    stats = Dict{String,Float64}()
    
    stats["mean_velocity_magnitude"] = mean(sqrt.(u.^2 + v.^2 + w.^2))
    stats["max_velocity_magnitude"] = maximum(sqrt.(u.^2 + v.^2 + w.^2))
    stats["mean_subgrid_viscosity"] = mean(nu_sgs)
    stats["max_subgrid_viscosity"] = maximum(nu_sgs)
    stats["mean_strain_rate"] = mean(strain_rate)
    stats["max_strain_rate"] = maximum(strain_rate)
    stats["turbulent_kinetic_energy_estimate"] = 0.5 * mean(u.^2 + v.^2 + w.^2)
    
    return stats
end

"""
    compute_les_timestep_2d(options, turbulence_model, x, y) -> Float64

Compute stable time step for 2D LES.
"""
function compute_les_timestep_2d(options::NSOptions, turbulence_model::TurbulenceModel, 
                                x::AbstractVector, y::AbstractVector)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    h_min = min(dx, dy)
    
    # Conservative time step for LES (subgrid viscosity can be large)
    max_velocity = 2.0  # Estimate
    dt_convection = 0.3 * h_min / max_velocity
    
    # Include potential subgrid viscosity contribution
    nu_total = options.nu + 0.1 * options.nu  # Estimated SGS viscosity
    dt_diffusion = 0.3 * h_min^2 / (2 * nu_total)
    
    return min(dt_convection, dt_diffusion)
end

"""
    compute_les_timestep_3d(options, turbulence_model, x, y, z) -> Float64

Compute stable time step for 3D LES.
"""
function compute_les_timestep_3d(options::NSOptions, turbulence_model::TurbulenceModel, 
                                x::AbstractVector, y::AbstractVector, z::AbstractVector)
    dx = length(x) > 1 ? abs(x[2] - x[1]) : 1.0
    dy = length(y) > 1 ? abs(y[2] - y[1]) : 1.0
    dz = length(z) > 1 ? abs(z[2] - z[1]) : 1.0
    h_min = min(dx, dy, dz)
    
    # More restrictive 3D time step
    max_velocity = 2.0
    dt_convection = 0.2 * h_min / max_velocity
    
    nu_total = options.nu + 0.1 * options.nu
    dt_diffusion = 0.2 * h_min^2 / (6 * nu_total)  # 3D diffusion stability
    
    return min(dt_convection, dt_diffusion)
end